On P. Hall's Generalisation of a Theorem of Frobenius

It is a well known theorem due to Frobenius that the number of solutions of the equation X n = 1 in a finite group G , is a multiple of the greatest common divisor ( n , g ) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equation X n = a where a is a fixed element of G , is a multiple of ( n , g a ), g a being the order of the centralizer Z(a) of a in G .